Strips such as steel strips, or strips made of other metals, can be subjected to a thickness reduction process e.g. by cold rolling or hot rolling in a mill. The work piece, i.e. the strip, is uncoiled from an uncoiler, processed in the mill, and coiled onto a coiler.
A mill comprises rolls with one set of rolls being arranged above the strip and another set of rolls being arranged below the strip when the strip passes through the mill. The mill is arranged to receive the strip between two work rolls forming a roll gap. The remaining rolls provide additional control and pressure to the work rolls, thereby controlling the roll gap profile and hence the flatness of the strip as it moves through the roll gap.
A cluster mill for example comprises a plurality of rolls stacked as layers above and below the work rolls. Backup rolls, i.e. the uppermost rolls of the rolls arranged above the roll gap and the lowermost rolls of the rolls arranged below the roll gap, may be segmented. Each roll segment may be moved in and out of the mill by means of crown actuators. The movements of the segmented rolls permeate through the cluster of rolls toward the work rolls for forming the strip moving through the roll gap. The remaining rolls of the cluster mill may also be actuated by means of their respective actuators. Bending actuators may for instance provide bending effects to a roll to which they are assigned and thereby change the profile of the roll gap. Side-shift rolls may have non-cylindrical shape which alters the roll gap profile by means of axial displacement of the side-shift rolls via side-shift actuators.
A uniform flatness across the width of the strip is typically desired as a non-uniform flatness may e.g. result in the manufacture of a strip having lower quality than a strip having an essentially uniform flatness profile. A strip having non-uniform flatness may for instance become buckled or partially corrugated. Non-uniform flatness may also cause strip breaks due to locally increased tension. Therefore, the flatness profile of the strip is measured, e.g. by measuring the force applied by the strip to a measurement roll, prior to the strip is coiled onto the coiler, wherein the measured flatness data is provided to a control system which controls the actuators of the mill for controlling the roll gap of the mill such that uniform flatness of the strip may be obtained. In order to control the actuators, the mill is generally modeled by means of a flatness response function for each of the actuators of the mill. These can for example be gathered as columns in a matrix, sometimes referred to as the mill matrix, Gm.
In a mill having a plurality of actuators, such as a cluster mill, one may have linear dependence among the flatness responses. This means that there may be actuator position combinations which do not affect the flatness of the strip because the combined flatness response provided by the actuators cancel the flatness effects provided by each individual actuator. For mills in which the above-described situation may arise, the corresponding mill matrix is said to be singular. In mathematical terms, a singular mill matrix does not have full rank, i.e. the mill matrix null space has a dimension greater than zero.
A classical control approach involves one control loop per actuator, with the flatness error vector projected to one value per control loop. For mills having a singular mill matrix this leads to such movement of the actuators that in some cases the flatness of the strip will not be affected, because the error projection allows all possible actuator position combinations. This corresponds to actuator movement in the null space of the mill matrix. Repeated disturbances will cause the actuators to drift along the directions which do not directly influence the flatness. There is also a risk that these actuator movements get far too large. These two cases of unwanted behavior may cause the actuators to saturate, but also cause unnecessary actuator load and wear.
In order to address this problem, the mill matrix Gm may be represented in the form of its singular value decomposition Gm=UΣVT. The singular values of Gm, which form the diagonal of Σ obtained from the singular value decomposition, provide information of the magnitude of the flatness response provided by each of the actuator position combinations, as defined by the column vectors of the orthonormal matrix V to flatness shapes as defined by the columns of the orthonormal matrix U. Moreover, the singular value decomposition provides information regarding actuator positions which do not directly influence the flatness profile of the roll gap, i.e. the null space.
By parameterizing the flatness error using the flatness response in the directions which do influence the flatness, and by mapping the controller outputs utilizing only those directions which do influence the flatness, movement of actuators in directions which do not influence the flatness may be blocked. Thus, actuator position combinations which do not affect the flatness profile of the roll gap will be avoided. By utilizing singular value decomposition to avoid combinations of the actuator positions which do not affect the flatness of the strip, not all degrees of freedom of control will be available for control in the sense that some combinations of actuator positions will not be allowed. Therefore control performance may suffer. Moreover, it may also be difficult to tune the separate control loops satisfyingly, since each control loop involves several actuators and therefore have more complex dynamics. EP2505276 addresses these problems by determining an adjusted flatness error based on the measured flatness error and weights for actuator positions which provide a flatness effect below a threshold value. Hence, in some situations the actuator position combinations which correspond to vectors in the null space of the model may be allowed. Thereby all possible actuator position combinations, i.e. all degrees of freedom of the control system which implements the method can be utilized.
Although singular value decomposition based flatness control has proved to be efficient, it is important to tune the process correctly in order to obtain successful flatness control.